Hyperfunction method for numerical integration and Fredholm integral equations of the second kind

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Hyperfunction Method for Numerical Integration
and Fredholm Integral Equations of the Second Kind
Hidenori Ogata
The University of Electro-Communications, Japan
13 July, 2017

Aim of this study
2 / 23
Hyperfunction theory (M. Sato, 1958)✓ ✏
• A theory of generalized functions based on complex function theory.
• A “hyperfunction” is expressed in terms of complex analytic functions.
hyperfunctions
= functions with singularities
pole
discontinuity
delta impluse, ...
←−
complex analytic function
easy to treat
numerically
✒ ✑
In this talk, we propose hyperfunction methods for
• numerical integration
• Fredholm integral equations of the second kind.

Contents
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1. Hyperfunction thoery
2. Hyperfunction method for numerical integration
3. Hyperfunction method for Fredholm integral equations
4. Summary

Contents
4 / 23
4. Summary

1. Hyperfunction theory
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Hyperfunction theory (M. Sato, 1958)✓ ✏
• hyperfunction on an interval I
. . . the difference between the values of a complex analytic funtion F(z) on I
f(x) = [F(z)] ≡ F(x + i0) − F(x − i0).
F(z) : deﬁning function of the hyperfunction f(x)
analytic in D I, where D is a complex neighborhood of I
✒ ✑
D
I
F(z)
=Re z
m z

1. Hyperfunctions: examples
6 / 23
Dirac’s delta function
δ(x) = −
1
2πi
1
x + i0
−
1
x − i0
.

1. Hyperfunctions: examples
6 / 23
Dirac’s delta function
δ(x) = −
1
2πi
1
x + i0
−
1
x − i0
.
O
D
a b
C
+ǫ
−ǫ
Suppose that φ(z) is analytic in D. By Cauchy’s integral formula,
φ(0) =
b
a
φ(x)δ(x)dx = −
1
2πi
b
a
φ(x)
1
x + i0
−
1
x − i0
dx.

1. Hyperfunction: examples
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Heaviside step function
H(x) =
1 ( x > 0 )
0 ( x < 0 )
= F(x + i0) − F(x − i0), F(z) = −
1
2πi
log(−z).
-0.4 -0.2 0 0.2 0.4 0.6 0.8 1Re z
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
Im z
-1
-0.5
0
0.5
1
Re F(z)
The real part of F(z) = −
1
2πi
log(−z).

1. Hyperfunction theory: integral
8 / 23
integral of a hyperfunction✓ ✏
f(x) = F(x + i0) − F(x − i0) : hyperfunction on an interval I
I
f(x)dx ≡ −
C
F(z)dz,
C : closed path encircling I in the positive sense and included in D
(F(z) is analytic in D I)
✒ ✑
D
C
I

1. Hyperfunction theory: integral
8 / 23
integral of a hyperfunction✓ ✏
f(x) = F(x + i0) − F(x − i0) : hyperfunction on an interval I
I
f(x)dx ≡ −
C
F(z)dz,
C : closed path encircling I in the positive sense and included in D
(F(z) is analytic in D I)
✒ ✑
D
C
I
I
f(x)dx =
I
[F(x + i0) − F(x − i0)] dx.

Contents
9 / 23
4. Summary

10 / 23
We consider an integral of the form
I
f(x)w(x)dx,
f(x) : analytic in D (I ⊂ D ⊂ C, )
w(x) : weight function.
D
I

10 / 23
I
f(x)w(x)dx,
D
I
We can regard the integrand as a hyperfunction.
✓ ✏
f(x)w(x)χI(x) = −
1
2πi
{f(x + i0)Ψ(x + i0) − f(x − i0)Ψ(x − i0)}
with χI(x) =
1 (x ∈ I)
0 (x ∈ I)
, Ψ(z) =
I
w(x)
z − x
dx.
✒ ✑

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I
f(x)w(x)dx,
D
C : z = ϕ(u)
I
We can regard the integrand as a hyperfunction.
✓ ✏
I
f(x)w(x)dx =
1
2πi C
f(z)Ψ(z)dz
=
1
2πi
τperiod
0
f(ϕ(τ))Ψ(ϕ(τ))ϕ′
(τ)dτ,
C : z = ϕ(τ) ( 0 ≦ τ ≦ τperiod ) periodic function (of period τperiod)
✒ ✑
Approximating the complex integral by the trapezoidal rule, we have ...

11 / 23
Hyperfunction method✓ ✏
I
f(x)w(x)dx ≃
h
2πi
N−1
k=0
f(ϕ(kh))Ψ(ϕ(kh))ϕ′
(kh),
with Ψ(z) =
b
a
w(x)
z − x
dx and h =
τperiod
N
.
✒ ✑
D
C : z = ϕ(τ), 0 ≦ τ ≦ τperiod
I

11 / 23
I
f(x)w(x)dx ≃
h
2πi
N−1
k=0
(kh),
with Ψ(z) =
b
a
w(x)
z − x
dx and h =
τperiod
N
.
✒ ✑
Ψ(z) for some typical weight functions w(x)
I w(x) Ψ(z)
(a, b) 1 log
z − a
z − b
∗
(0, 1) xα−1
(1 − x)β−1
B(α, β)z−1
F(α, 1; α + β; z−1
)∗∗
( α, β > 0 )
∗ log z is the branch s.t. −π ≦ arg z < π.
∗∗ F(α, 1; α + β; z−1
) can be easily evaluated using a continued fraction.

11 / 23
I
f(x)w(x)dx ≃
h
2πi
N−1
k=0
(kh),
with Ψ(z) =
b
a
w(x)
z − x
dx and h =
τperiod
N
.
✒ ✑
If f(z) is real-valued on R, we can reduce the number of sampling points N by half
using the reﬂection principle.

2. Numerical integration: example
13 / 23
✓ ✏
1
0
ex
xα−1
(1 − x)β−1
dx = B(α, β)F(α; α + β; 1) ( α, β > 0 ).
✒ ✑
We computed this integral by
• hyperfunction method (with N reduction),
• DE formula (efﬁcient for integrals with end-point singularities)
• Gauss-Jacobi formula
• C++ program, double precision
• complex integral path for the hyperfunction method (an ellipse)
z = ϕ(τ) =
1
2
+
1
4
ρ +
1
ρ
cos τ +
i
4
ρ −
1
ρ
sin τ ( ρ = 10 )
= 0.5 + 2.575 cos τ + i2.425 sin τ.

14 / 23
-16
-14
-12
-10
-8
-6
-4
-2
0
0 10 20 30 40 50 60
log10(error)
N
hyperfunction
hyperfunction
Gauss-Jacobi
Gauss-Jacobi
DE
DE
-16
-14
-12
-10
-8
-6
-4
-2
0
0 20 40 60 80 100 120
log10(error)
N
hyperfunction
hyperfunction
Gauss-Jacobi
DE
DE
α = β = 0.5 α = β = 10−4
(very strong singularities)
The errors of the hyperfunction method, Gauss-Jacobi formula and the DE formula
hyperfunction Gauss-Jacobi DE
α = β = 0.5 O(0.025N
) O((8.2 × 10−4
)N
) O(0.36N
)
α = β = 10−4
O(0.029N
) — O(0.70N
)

14 / 23
-16
-14
-12
-10
-8
-6
-4
-2
0
0 10 20 30 40 50 60
log10(error)
N
hyperfunction
hyperfunction
Gauss-Jacobi
Gauss-Jacobi
DE
DE
-16
-14
-12
-10
-8
-6
-4
-2
0
0 20 40 60 80 100 120
log10(error)
N
hyperfunction
hyperfunction
Gauss-Jacobi
DE
DE
α = β = 0.5 α = β = 10−4
(very strong singularities)
The hyperfunction method converges geometricaly,
and its performance is not affected by the end-point singularities.

Contents
15 / 23
4. Summary

3. Hyperfunction method for integral equations
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Fredholm integral equation for unknown u(x)✓ ✏
λu(x) −
b
a
K(x, ξ)u(ξ)w(ξ)dξ = g(x),
w(ξ) : weight function, K(x, ξ), g(x), λ(= 0) : given.
✒ ✑
We apply the hyperfunction method to this integral equation.

17 / 23
λu(x) −
b
a
K(x, ξ)u(ξ)w(ξ)dξ = g(x).
(Assumption)
• g(z) : analytic in D except for
a ﬁnite number of poles at a1, . . . , aK
• K(z, ζ) : analytic function in D w.r.t. z and ζ D
a b
ak

17 / 23
λu(x) −
b
a
K(x, ξ)u(ξ)w(ξ)dξ = g(x).
(Assumption)
• g(z) : analytic in D except for
a ﬁnite number of poles at a1, . . . , aK
• K(z, ζ) : analytic function in D w.r.t. z and ζ D
a b
ak
ua(z) ≡ u(z) − λ−1
g(z) is analytic in D.
ua(x) satisﬁes the integral equation
✓ ✏
λua(x) −
b
a
K(x, ξ)ua(ξ)w(ξ)dξ =
1
λ
b
a
K(x, ξ)g(ξ)w(ξ)dξ.
✒ ✑
1. We discretize the integral equation for ua(x) by the hyperfunction method.
2. We solve the discretized equation by the collocation method.

3. Integral equations: Collocation equation
18 / 23
h
2πi
N
k=1
λ
ϕ(kh) − zi
− K(zi, ϕ(kh))Ψ(ϕ(kh)) ϕ′
(ϕ(kh))ua(ϕ(kh))
=
1
2πiλ C
K(zi, ζ)g(ζ)Ψ(ζ)dζ−
1
λ
N
k=1
Res(K(zi, ·)Ψg, ak) (i = 1, . . . , N),
where
C : z = ϕ(τ) ( 0 ≦ τ ≦ τperiod ) closed path encircling [a, b],
periodic function (period τperiod)
z1, . . . , zN : the collocation points inside C, h = τperiod/N.
The collocation equation
... a system of linear equations for ua(ϕ(kh))
( k = 1, . . . , N ). a b
ak
C : z = ϕ(τ)
D
zi

3. Integral equations: Collocation equation
18 / 23
h
2πi
N
k=1
λ
ϕ(kh) − zi
− K(zi, ϕ(kh))Ψ(ϕ(kh)) ϕ′
(ϕ(kh))ua(ϕ(kh))
=
1
2πiλ C
K(zi, ζ)g(ζ)Ψ(ζ)dζ−
1
λ
N
k=1
Res(K(zi, ·)Ψg, ak) (i = 1, . . . , N),
where
C : z = ϕ(τ) ( 0 ≦ τ ≦ τperiod ) closed path encircling [a, b],
periodic function (period τperiod)
z1, . . . , zN : the collocation points inside C, h = τperiod/N.
The approximate solution u(z) is given by
u(z) =
1
2πi C
ua(ζ)
ζ − z
dζ + g(z)
≃
h
2πi
N
j=1
ua(ϕ(kh))
ϕ(kh) − z
ϕ′
(kh) + g(z).
a b
ak
C : z = ϕ(τ)
D
zi

3. Integral equations: example
19 / 23
✓ ✏
u(x) +
1
0
(x − ξ)u(ξ)ξα−1
(1 − ξ)β−1
dξ = g(x),
g(x) =
1
1 + x2
+ B(α, β) Re{F(α, 1; α + β; i)}x
− B(α + 1, β) Re{F(α + 1, 1; α + β + 1; i)} ( α = β = 0.5, 10−4
).
✒ ✑
We solved the integral equation by the hyperfunction method, DE-Nystr¨om method and
Gauss-Jacobi-Nystr¨om method.
• complex integral path
C : z = ϕ(τ) =
1
2
+
1
4
ρ +
1
ρ
cos τ +
i
4
ρ −
1
ρ
sin τ ( ρ = 200 )
• collocation points zi = ϕcol
2π(i − 1)
N
( i = 1, . . . , N )
ϕc(τ) =
1
2
+
1
4
ρc +
1
ρc
cos τ +
i
4
ρc −
1
ρc
sin τ ( 1 < ρc < ρ ).

3. Integral equations: example (α = β = 0.5)
20 / 23
-60
-50
-40
-30
-20
-10
0
0 20 40 60 80 100
log10(error)
N
rhoc=1.2
rhoc=2.0
rhoc=4.0
rhoc=6.0
rhoc=8.0
DE
Gauss-Jacobi
0
20
40
60
80
100
0 20 40 60 80 100
log10(cond)
N
rhoc=1.2
rhoc=2.0
rhoc=4.0
rhoc=6.0
rhoc=8.0
DE
Gauss-Jacobi
error ǫN condition number κN of
the collocation equation
(rhoc = ρc)
ρc/ρ 0.006 0.01 0.02 0.03 0.04
ǫN O(0.0058N
) O(0.010N
) O(0.020N
) O(0.030N
) O(0.040N
)
κN O(160N
) O(97N
) O(48N
) O(32N
) O(23N
)

3. Integral equations: example (α = β = 0.5)
20 / 23
-60
-50
-40
-30
-20
-10
0
0 20 40 60 80 100
log10(error)
N
rhoc=1.2
rhoc=2.0
rhoc=4.0
rhoc=6.0
rhoc=8.0
DE
Gauss-Jacobi
0
20
40
60
80
100
0 20 40 60 80 100
log10(cond)
N
rhoc=1.2
rhoc=2.0
rhoc=4.0
rhoc=6.0
rhoc=8.0
DE
Gauss-Jacobi
(rhoc = ρc)
ρc/ρ 0.006 0.01 0.02 0.03 0.04
ǫN O(0.0058N
) O(0.010N
) O(0.020N
) O(0.030N
) O(0.040N
)
κN O(160N
) O(97N
) O(48N
) O(32N
) O(23N
)
• error ǫN = O[(ρc/ρ)N
], cond. number κN = O[(ρ/ρc)N
].
• converges faster than the DE-Nystr¨om method.
• The linear system of the collocation equation is very ill-conditioned.

3. Integral equations: example (α = β = 10−4
)
21 / 23
-60
-50
-40
-30
-20
-10
0
0 20 40 60 80 100
log10(error)
N
rhoc=1.2
rhoc=2.0
rhoc=4.0
rhoc=6.0
rhoc=8.0
DE
Gauss-Jacobi
0
20
40
60
80
100
0 20 40 60 80 100
log10(cond)
N
rhoc=1.2
rhoc=2.0
rhoc=4.0
rhoc=6.0
rhoc=8.0
DE
Gauss-Jacobi
(rhoc = ρc)
ρcol/ρ 0.006 0.01 0.02 0.03 0.04
ǫN O(0.0058N
) O(0.010N
) O(0.020N
) O(0.030N
) O(0.040N
)
κN O(160N
) O(97N
) O(48N
) O(32N
) O(24N
)

3. Integral equations: example (α = β = 10−4
)
21 / 23
-60
-50
-40
-30
-20
-10
0
0 20 40 60 80 100
log10(error)
N
rhoc=1.2
rhoc=2.0
rhoc=4.0
rhoc=6.0
rhoc=8.0
DE
Gauss-Jacobi
0
20
40
60
80
100
0 20 40 60 80 100
log10(cond)
N
rhoc=1.2
rhoc=2.0
rhoc=4.0
rhoc=6.0
rhoc=8.0
DE
Gauss-Jacobi
(rhoc = ρc)
ρcol/ρ 0.006 0.01 0.02 0.03 0.04
ǫN O(0.0058N
) O(0.010N
) O(0.020N
) O(0.030N
) O(0.040N
)
κN O(160N
) O(97N
) O(48N
) O(32N
) O(24N
)
• error ǫN = O[(ρcol/ρ)N
], cond. number κN = O[(ρ/ρcol)N
].
• The DE-Nystr¨om method does not work if the end-point singularities are
very strong.

Contents
22 / 23
4. Summary

4. Summary
23 / 23
• We applied hyperfunction theory to numerical integration and Fredholm integral
equations of the second kind.
◦ Hyperfunction theory: a generalized function theory where a “hyperfunction” is
expressed in terms of complex analytic functions.
◦ A hyperfunction integral is given by a complex loop integral, which is evaluated
numerically in the hyperfunction method.
• Hyperfunction method
◦ (Theoretical error estimate) geometric convergence
◦ (Numerical examples) efﬁciency for problems with strong end-point singularities
◦ Integral equation: The linear system of the collocation equation is
very ill-conditioned.
• Problems for future study
◦ Volterra integral equations.
◦ theoretical error estimate.

4. Summary
23 / 23
• We applied hyperfunction theory to numerical integration and Fredholm integral
equations of the second kind.
◦ Hyperfunction theory: a generalized function theory where a “hyperfunction” is
expressed in terms of complex analytic functions.
◦ A hyperfunction integral is given by a complex loop integral, which is evaluated
numerically in the hyperfunction method.
• Hyperfunction method
◦ (Theoretical error estimate) geometric convergence
◦ (Numerical examples) efﬁciency for problems with strong end-point singularities
◦ Integral equation: The linear system of the collocation equation is
very ill-conditioned.
• Problems for future study
◦ Volterra integral equations.
◦ theoretical error estimate.
Thank you!

Hyperfunction method for numerical integration and Fredholm integral equations of the second kind

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